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**Quenching measurements of complex chemical dynamics
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The research group has developed the theoretical and experimental basis of a method of analysing oscillatory chemical reactions (the quenching method). Quenching measurements for complex chemical systems in a CSTR operated close to a Hopf bifurcation is an effective tool for determining the essential species and elementary reactions from a comprehensive kinetic mechanism for the system. A reduced set of species and reactions obtained in this way may be a good starting point for a detailed understanding of bifurcations and chaotic behavior, as the operating conditions for a Hopf bifurcation is often not far from the operating conditions for chaotic oscillations. Quenching results on the Belousov-Zhabotinsky (BZ) reaction at a Hopf bifurcation have been compared with several established kinetic models. Qualitative or semiquantitative agreement has been obtained, but in a few cases considerable disagreements between the experiments and the predictions from existing models show the need for substantial changes of the models. We cooperate with R. J. Field from Montana on these problems.

A supercritical Hopf bifurcation has been discovered in the Briggs-Rauscher reaction, and quenching parameters have been measured. An interesting paradox unexplainable by the established models has turned up.

A model for the methylene blue-sulfide reaction has been investigated
by numerical methods and an operating point on the thermodynamic branch
corresponding to a supercritical Hopf bifurcation has been discovered.
This behavior has not been confirmed experimentally, but for very large
flow rates with influx of hydrogenperoxide nearly harmonic oscillations
have recently been observed suggesting the presence of a supercritical
Hopf bifurcation.
-0.1truecm *F. Hynne, P. Graae Sørensen, V. Vukojevic, K. Nielsen, M. Ipsen.
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**Analytic methods for systematic optimization of models of
oscillatory reactions.
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Analytical methods of assessing and optimizing networks of oscillatory chemical reactions are being developed. They exploit the close relationship that exists between quenching data of a system and the kinetics of its reactions. Two methods are currently being implemented and applied. Both can be used directly without integrating the kinetic equations.

A graphical method is based on an eigenvalue property of
quenching data. For a given reaction network, it provides necessary
conditions for the consistency of models with the experimental data.
The other method depends on a unique property of a Hopf bifurcation:
it uses an explicit representation of stationary states in terms
of 'extreme currents'. It provides a way to systematically search
all possible Hopf bifurcations of a network in an optimization,
regardless of rate constants and stationary concentrations.
-0.1truecm *F. Hynne, P. Graae Sørensen, T. Møller.
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**Chaos in oscillatory chemical reactions.
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The geometry of fully developed Feigenbaum chaos in a chemical
system is best studied and described by determination of the
(infinite) set of unstable limit cycles and the associated
manifolds.
Chemical chaos is extremely difficult to
handle experimentally, and much work on it has been made under conditions that
are not sufficiently well controlled, either from a chemical or from
a dynamical point of view. Even early
stages of the period-doubling sequence present a
considerable challenge.
Several types of chaos, in particular Feigenbaum chaos and
homoclinic chaos, are being investigated experimentally in
systems driven by periodic perturbations.
-0.1truecm *P. Graae Sørensen, K. Nielsen, F. Hynne.
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**Turing structures: Self-organization through bifurcations in
biochemical control systems with diffusion
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Spontaneous creation of stable concentration patterns (morphogenetic fields)
is possible in autocatalytic biochemical control systems. Such patterns
may emerge from the homogeneous state if the activity of certain enzyme
passes over a critical threshold or simply if the size of the embryo
grows over a critical value. The underlying mathematics describing the
system shows profound analogies to models of transition to turbulence
in fluid dynamics. The patterns arise through bifurcation in the nonlinear
partial differential equation describing the autocatalytic
reaction-diffusion system. At a NATO meeting in Brussels Dec 2-5, 1991
G. Nicolis pointed out that such Turing systems may be easier to analyze
than hydrodynamic systems and recommended close cooperation in the two fields.
H. Swinney (Austin, Texas), who is an expert on the transition to turbulence,
presented experiments on chemically realized Turing structures. In
collaboration with P. Graae Sørensen we have established how Turing systems
may adapt to size in growing embryos and how the effective diffusion
constants in biosystems may vary sufficiently. This work has been used
by the Bordeaux group in its recent discovery of experimentally realized Turing
structures in a gel reactor. In the last decade we have pioneered the simulation
of Turing structures in 3 curvilinear coordinates which make direct comparisons
to biological systems possible. A number of otherwise unexplainable patterns
have been interpreted by us as Turing patterns, and a possible link between
such patterns and cell division (mitosis and cytokinesis) and segmentation
in *Drosophila* has been suggested. Currently a model with 16
interacting species is used to simulate a combined model of Turing
morphogens, gap- and primary pair-rule genes in 3D elliptical cylinder
coordinates with a fast supercomputer code (> 1300 MFLOPS on CRAY C92).
This code is used by the national computer center to benchmark current
supercomputers.
-0.1truecm *A. Hunding, [T. Lacalli
(Saskatchewan), J. Boissonade (Bordeaux)]
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**Hopf bifurcation in biochemical systems.
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Enzymatic reactions with oscillatory properties has been investigated
by traditional kinetic methods for many years. We expect that
quenching measurements may contribute to new theoretical
understanding of this very important class of systems. Experiments
with such
systems require great experience in biochemical technique, and we are
planning a collaboration with H.G.Busse at the University of Kiel
on investigations of glycolysis.
-0.1truecm *K. Nielsen, F. Hynne, P. Graae Sørensen, [H. G. Busse (Kiel)].
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**Reinforced dynamics.
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This study concerns relaxation oscillators, coupled by a
reinforced dynamics, where the coupling strength between two
neighboring oscillators increases when firing at one leads to firing
at the other. We consider both the case, where the system has a
constant drive, and the case, where the drive is a function of the
outcome. Our goal is (i) to explore the patterns formed, and their
stability, when the system is driven; (ii) to study the systems
ability to perform well-defined tasks.
-0.1truecm *D. Stassinopoulos, P. Alstrøm.
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**Turing Structures in the CIMA Reaction
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Using the Chlorite-Iodide-Malonic Acid (CIMA) reaction, in 1990
de Kepper and his group in Bordeaux finally succeeded in
attaining stationary spatial structures. We have studied the one- and
two-dimensional bifurcation diagrams of a model of the CIMA
reaction. Because of multistability and pinning effects a great variety
of localized structures may be produced, such as for instance,
islands of hexagons surrounded by homogeneous steady state. We
have determined the speed at which fronts between such structures
propagate, and we have shown how one- and two-dimensional spirals
with Turing induced cores can arise.
-0.1truecm *(P. Borckmans), (G. Dewel), O. Jensen, E. Mosekilde, and
V.O. Pannbacker
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Mon Feb 6 10:24:27 MET 1995